守恒律双曲系统的统计解：数值逼近
Statistical solutions of hyperbolic systems of conservation laws: numerical approximation
AI总结：
本文提出结合高分辨率有限体积法与蒙特卡洛采样的数值算法，用于逼近守恒律双曲系统的统计解，并证明其收敛性及数值实验验证。
摘要：
统计解是积分函数空间上的时间参数化概率测度，最近被提出作为多维守恒律双曲系统全局解和不确定性量化的框架。通过将高分辨率有限体积方法与蒙特卡洛采样过程相结合，我们提出了一种逼近统计解的数值算法。在有限体积方法可验证的假设下，我们证明了由所提算法生成的逼近在适当的拓扑下收敛于一个统计解。还给出了说明收敛理论并揭示统计解有趣性质的数值实验。
英文摘要：
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions, are also presented.
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